linear programming

[mhernandez24]

I know how to do the problem. I'm just having trouble figuring out the inequalities for the graph.

An airline with two types of airplanes, A and B, has contracted with a tour group to provide accommodations for a minimum of each of 2000 first class, 1500 tourist, and 2400 economy-class passengers. Airplane A costs $12,000 per mile to operate and can accommodate 40 first-class, 40-tourist and 120 economy-class passengers, whereas airplane B costs $10,000 per mile to operate and can accommodate 80 first-class, 30 tourist, and 40 economy-class passengers. How many of each type of airplane should be used to minimize the operating cost? What is the minimum cost?

 

[mmm4444bot]

just having trouble figuring out the inequalities

Hi M,

In future requests for help, please show what you've done so far or explain your thinking; otherwise, I type a bunch of stuff that you already know because I can't determine where to start with you. I mean, did you pick symbols, yet?

two types of airplanes, A and B ... How many of each type of airplane

This given information tells you that you're seeking two specific numbers. It also suggests symbols, to represent these two unknown numbers.

So, here's the first step: Provide your symbol definitions

Let A = the number of type-A planes needed

Let B = the number of type-B planes needed

Right away, we can write two inequalities:

A > 0

B > 0

Within the rest of the given information, the "stuff" for writing the remaining inequalities all pertains to plane counts, passenger-counts, and traveling-class minimum requirements. I find it's helpful to make a chart because seeing the information organized gels the relationships, in my mind.

From my chart, I "see" how to write expressions for the following.

The number of first-class passengers flying in all planes combined: 40A + 80B

The number of tourist-class passengers flying in all planes combined: 40A + 30B

I'll leave the expression representing the total economy-class passenger-count, in all planes combined, for you to write.

The exercise tells us that the given totals (for each of the three traveling classes) are minimums ("accommodations for a minimum of each of 2000 first class", etc… ). Therefore, the actual number of passengers in any traveling class might be higher than the givens, so we use the "greater than or equal to" symbol >=.

40A + 80B >= 2000

Can you write the remaining two inequalities (i.e., for the tourist and the economy seats) ?


Airplane A costs $12,000 per mile to operate

airplane B costs $10,000 per mile to operate

This is the information for the objective expression; I mean, the expression that you're trying to minimize.

But, of course, you know how this part works because you already know how to "do" the problem and just need help with the inequalities.

I welcome specific questions.

Cheers ~ Mark

MY EDITS: Fixed two typos, changing > to >

 

[mhernandez24]

Thank you for your help. i solved the problem.

 

[mmm4444bot]

You are welcome.

I get: 30 type-A planes and 10 type-B planes will minimize combined-operating cost.

I also notice that my inequalities A>0 and B>0 contain typographical errors, in my previous post, as zero should be included. I'll fix that.

Cheers